29 problems

Beilinson–Bloch conjecture. The Abel–Jacobi map is injective for varieties defined over number fields; consequently, every such cycle that is trivial under the Abel–Jacobi map is t…

Let $k$ be a number field, let $\mathbb{V}(k)$ denote the relevant class of varieties over $k$, and let $X\in\mathbb{V}(k)$. The $l$-adic Abel–Jacobi map is the map from the releva…

Let $X$ be the smooth projective complex algebraic variety in (0.4), of dimension $n$, and let $Y$ be an intersection of $n-2$ general hyperplane sections of $X$. Assume that $Y$ h…

Let $X\subseteq \mathbb P^N$ be a smooth projective variety of dimension $n$, let $Y=X\cap H$ be a smooth hyperplane section, let $U\subseteq (\mathbb P^N)^*$ be the open set of sm…

Voisin–Beilinson conjectures. For $p=2$, $\text{CH}_{\text{ind}}^{2}(X,1)_{{\Bbb Q}}$ should be countable, and the map in (4.5.2) should be surjective.

Let $X$ and $S$ be as in (0.6), and suppose $\d7eta \in \text{CH}^{2}(X,1)_{{\Bbb Q}}$ restricts to zero on the relevant fibers. The higher Abel–Jacobi map … is injective for any f…

Let $X$ be a projective nonsingular variety over $\bold C$. Let $F^2_{\cal M}{\mathrm{CH}}^r(X)$ denote the motivic filtration, and let … be the Abel-Jacobi map. Motivic-filtration…

Let $r\geq 2$ be an integer. For a nonsingular projective variety $X$ over $\overline{\bold Q}$, let … be the Abel-Jacobi map. If $X_{\bold C}=X\otimes_{\overline{\bold Q}}\bold C$…

Termination conjecture for the motivic filtration. The filtration terminates, meaning

Let $X$ be a smooth projective variety of dimension $g$, and let $V\subset H^m(X,\mathbb{Q})$ be a $\mathbb{Q}$-Hodge substructure of level at most $m-2p$. A nonsingular projective…

Extension-class conjecture. For any $c\in(H^{2p}(X_{\overline{k}},{\mathbb Q}_l(p))^D)^{G_k}$, the extension class $e(c)$ belongs to the image of (0.3).

Let $k$ be finitely generated over its prime subfield, let $l$ be invertible in $k$, and let $\mathcal Y$ be a regular proper model of $k$ over $\mathbb Z$. Let $X$ be a smooth pro…

Let $\nu:{\cal B}\to\widetilde{{\cal J}}$ be the inclusion of the total space of a family of abelian subvarieties, and let $\widetilde{{\cal X}}\to\widetilde{{\cal M}}$ be the corr…

Griffiths's conjecture on incidentally trivial cycles. For every $p\in\{1,\ldots,d\}$ and every $z\in I^p(X)$, there is a positive integer $N$ such that

Eskandari–Murty's conjecture. The cycle

Jacobi-type theorem. There is some positive integer $k$ such that the topological Abel--Jacobi mapping

Strong Tube Theorem. The image of the topological Abel--Jacobi homomorphism

One-elementary-vanishing-cycle generation conjecture. Only one elementary vanishing cycle is enough to generate $H_{2n-1}^{\operatorname{van}}(X,\mathbb{Q})$.

Let $\mathcal{L}$ be an A-model object, let $\mathcal{H}^{\text{even}}$ be the even cohomology space, and let … be the intermediate Jacobian. Quantum Abel–Jacobi conjecture. A syst…

Let $V$ be a smooth proper variety over a number field, let $r$ be any codimension, and let $\ell$ be a prime. Consider the $\ell$-adic Abel–Jacobi map from the relevant codimensio…

Let $X$ be a projective hyper-Kähler manifold of generalized Kummer type, and let $J^3(X)$ be its intermediate Jacobian. The degree-three cohomology is of Hodge level $1$, of type…

Let $W$ be a smooth projective variety over $\overline{\mathbb{Q}}$, and let $r$ and $m$ be integers for which the displayed higher Chow group is defined. Let … be the generalized…

Let $W$ be a smooth projective variety over $\overline{\mathbb{Q}}$, and let $r\geq 0$ be an integer. Let $J(H^{2r-1}(W(\mathbb{C}),\mathbb{Q}(r)))$ denote the relevant intermediat…

Let $Y_0$ be the smooth projective surface appearing as the factor of the special fiber $X_0=Y_0\times Q_0$, and let $H^2_{\rm tr}(Y_0,\mathbb{Q}(2))$ denote its transcendental coh…

Let $K$ be the field in the construction, $I$ an ideal, $H_I$ the associated extension, $G_I$ its relevant Galois group, $\tilde v_1$ a place of $H_I$ above $v_1$, and…